User blog:Syst3ms/Chained Array Notation
Hi, this is my new attempt at making a powerful notation, and this time I think I got it. Stage 1 : Bare-bones (BBCAN) : This has only one rule. a0 = a^a The limit of this notation is n0...0 with n 0's Stage 2 : Single Entry (SECAN) : with a b's (evaluated from left to right) n1 = n0...0 with n 0's Now it's pretty easy to see that this is basically Steinhaus-Moser notation, so 22 is the Mega. ab in CAN is equal to ab+3 in Steinhaus-Moser notation. The limit of this system in the FGH is . Now let's add more entries. Stage 3 : Linear (LCAN) : Let # be any string of numbers a0 = a^a a\#,0 = a\# ab+1,\# = ab,\#b,\#\cdotsb,\# \text{ with a b,#'s} a0,\ldots,0,b+1,\# = aa,\ldots,a,b,\# Since n0,1 = nn, 0,1 has strength n1,1 = n0,1...0,1, so 1,1 has strength Hence, na,1 has strength Therefore, 641,1 > Graham's number 0,2 has strength a,2 has strength 0,3 has strength a,b has strength The limit of LCAN is Stage 4 : Nested (NCAN) : Now we can nest brackets. A valid array would be [1,0,1,3] The previous rules still apply. If none of the rules apply, start the following process, starting at the first entry : 1) If the entry is a 0, jump to the next entry. 2) Otherwise, it is an array : 2a) First, remove trailing zeroes, if any 2b) If the entry is 0, replace it with the base. 2c) If the entry is of the form b+1,#, then decrease b by 1 and make a copies of it, where a is the base. 2d) If the entry is of the form 0,...,0,b+1,#, replace it with a,...,a,b,#, where a is the base. 2e) Otherwise, start the process inside the entry. with n n's Stage 5 : Super-nested (SNCAN) : This time we introduce more powerful brackets, the first of which being {} Then we can have <>, etc. To generalize, we can have []_2 be {}, []_3 be <>, etc. []_0 are null brackets, which are just a fancy name for the absence of brackets. All previous rules still apply. Inside the base rules, we add the following rule after the first one : a0_{b+1} = a[ [ \cdots a_b \cdots ]_b ]_b This rule replaces step 2b inside of the process. Since we replaced step 2b altogether, what happens when we do n[￸0] ? Well, it is equal to n((..(n)...)) where () are null brackets. Since those mark an absence of brackets, it is really equal to nn with n brackets with n brackets n{0,1} = n{n} We can even nest multiple types of brackets. This means that [{1},3,1,{0},1,1_4] is a perfectly valid array. The limit of this notation is n0_n Stage 6 : Linear-nested (LNCAN) : We can now use regular arrays inside of bracket subscripts. We change rule 2b to the following : 2b) If the expression is of the form , then it is equal to Otherwise : First, remove trailing zeroes in the subscript, if any. 2b.1) If the subscript is of the form b+1, \# , then the expression is equal to a0_{b,\#}\cdots0_{b,\#} with a 0's 2b.2) If it is of the form 0,\ldots,0,b+1,\# , then the expression is equal to a0_{a,\ldots,a,b,\#} Now, We can now use the full power of linear arrays. Stage 7 : Nested-nested (NNCAN): This extension allows for the entire power of nested arrays. The process stays pretty much the same, except that if the subscript rules don't apply, we start the process inside it. n0_{1,0,1} = n0_{1,n} Stage 8 : Powernested (PnCAN) : Now we can use nested-nested arrays in subscripts. This makes the limit of the notation introduced so far with n subscripts. 20_{\{1\}} = 20_{\{0\}}0_{\{0\}} = 20_{2}0_{\{0\}} = \ldots Category:Blog posts